Method and a device for generating an equal-tempered tone scale in musical instruments



July 21, 1970 K. ca. MALMFORS 3,520,932

METHOD AND A DEVICE FOR GENERATING AN EQUAL|-TEMPERED TONE SCALE IN MUSICAL INSTRUMENTS Filed June 29, 1967 2 Sheets-Sheet 1 n n-S D1 5 A B C A scn an M 02 P I i n 3 ST FF 1P5 FF Sc l I I ZE- 1 1 R 1 I l I R Adn n 1 INVENTOR. KARL G. MALMFORS AG EN'T July 21, 1970 K. G. MALMFORS 2 METHOD AND A DEVICE FOR GENERATING AN EQUAL-TEMPERED TONE SCALE IN MUSICAL INSTRUMENTS Filed June 29, 196'?- 2 Sheets-Sheet 2 WWW 1mm d o O/QT/K/T/K/Z/Z/C/ n A B C INVENTOR. KARL G. MALM FORS a Q l nssrvrk United States Patent METHOD AND A DEVICE FOR GENERATING AN EQUAL-TEMPERED TONE SCALE IN MUSICAL INSTRUMENTS Karl G. Malmfors, Viggbyholm, Sweden, assignor, by mesne assignments, to U.S. Philips Corporation, New York, N.Y., a corporation of Delaware Filed June 29, 1967, Ser. No. 649,984

Claims priority, application Sweden, June 30, 1966,

6 Int. Cl. H031 3/04; G10h 5/06 U.S. Cl. 841.01 Claims ABSTRACT OF THE DISCLOSURE An electronic musical instrument employing precise tone control by use of a number of voltage controlled oscillators. All but the lowest frequency oscillator employ automatic frequency control and consist of a Schmltt trigger with an RC feedback. Each of the frequency controlled stages derive their control from two other oscillators in accordance with a pre-set relationship.

The invention relates to a method for generating a tone scale preferably a substantially equal-tempered tone scale in musical instruments, preferably electronic musical instruments.

According to a known method from a number of free running oscillators which number is equal to the number of tones per octave, and which oscillators are each tuned to a different pitch tones of the lower octaves are derived by means of sealers of two.

It is clear that in case of detuning of one or more of these oscillators all tones derived therefrom detune as well. This is obviated with the method according to the invention in that each frequency is derived by means of frequency comparing and dividing or multiplicating means from two other frequencies and l m part of the frequency of each tone is equal to the frequency difference of two other tones of the tone scale, in which m is an integer, at least one of the frequencies being constant or adjustable independently of the remaining frequencies.

As result the frequencies of all tones in the octave are fixed with respect to the frequency of the free running oscillator so that the mutual frequency ratio which determines the right tuning of the instrument remains constant.

If in the system of 31 tones per octave in which the intervals between two adjacent tones is equal to x /fit is chosen to be equal to which value deviates only 1.3 10- of the right value 0.0270405 which deviation cannot be perceived by the ear. In the above example the frequency of f is normalized to 1.

For obtaining an equal-tempered twelve-tone scale the frequencies of groups of three tones satisfy the equation in which n is the ordinal number of the tone in the tone scale.

According to the definition an octave in a equally tempered scale is divided in twelve intervals such that the ratio between adjacent frequencies is constant equal to ice If the lowest one of the twelve frequencies f is assumed to be 1 the remaining frequencies will "be as follows:

TABLE 1' 1 :1.000 000 f =1.414 214 :1.059 463 7,:1498 307 ,=1.122 462 ,=1.5s7 401 f =1.189 207 f16=1.681 793 ,=1.259 921 13 :1181 797 f =1.334 s f =1.887 749 An investigation of the frequency values in Table l valid for one octave shows that the difference between the frequencies i and f is very closely equal to A of the basic frequency f As the frequencies form a geometric series a corresponding relation is valid also for other adjacent frequencies in the scale, which relation can be written more generally as:

where f is the frequency of the tone and n is the number of the tone in the octave as counted from the lowest tone.

Assume that one of the frequencies in the octave is given, suitably the basic tone f which is normalized to l.

The remaining frequencies can then be calculated in accordance with the above approximative relation between the frequencies. The following system of equations is obtained.

f1z= (f1of9) f11= (fa-f8) f10= (fsf7) 13 (fv-fe) f8= (fsf5) f7= (f5f4) Solution of the system of equations gives the following values on the frequencies.

TABLE 2 f =l.O59 460 -3.1 f =1.498 275 21.3 7 21.122 455 -6.2 f =1.587 366 22.4 13:1.189 196 -9.3 fw= .681 753 -23.5 f =1.259 905 -12.4 f =1.78l s09 +6.2 7,:1334 819 -15.5 f1z=1.887 754 +3.1 1414 187 -18.4

The right column in the Table 2 shows the deviation is parts per million between the frequencies obtained by use of the approximative relationship and the ideal frequencies given in Table 1. The largest deviation occurs as shown in the ratio between and f but also this deviation is less than 30 parts per million.

According to another method according to the invention for obtaining an equal-tempered twelve-tone scale the frequencies of groups of three tones satisfy the equation in which n is the ordinal number of the tone in the tone scale.

In this case the deviation from the ideal frequencies is somewhat larger but the scale is still usefull in every respect.

In a device for effectuating the method it is suitable to use twelve oscillator circuits adaptedto produce each a tone in the octave, which oscillator circuits have a basic frequency setting which approximately coincides with the appropriate frequency for the respective oscillator circuit and which oscillator circuits are furthermore frequency controlled by means of signals derived from two other oscillators in accordance with the given relationship, at least one oscillator frequency being set independently of remaining frequencies.

The maintenance of the given relation between the frequencies by frequency control of the oscillators may in a principle be effected in different Ways. If in Equation 1 the frequencies ;f,, and f are regarded as given frequencies i can be generated by multiplying by 20 the beat frequency obtained from superposition of f,, and f It was found, however, that this method leads to practical difficulties associated with the high multiplication factor.

A more suitable device for the frequency control is obtained if the oscillator frequency i is instead divided by 20 and the so-produced frequency is compared with the beat frequency of the two other oscillator frequencies according to the given relationship, a control signal produced by the comparison being applied to one of the oscillators for maintaining the said relation between the frequencies by negative feed back in the regulation circuit so formed.

In this way frequency multiplication is avoided and instead frequency division is to be performed, which is an easier problem in practice. The division can suitably be effected in three steps, first, two frequency divisions by two, for example by means of bistable flip-flops, and thereafter a division by five in a suitable device, the frequencies obtained by the said divisions by two being utilized for generating the corresponding tones in adjacent lower octaves.

If both the independent oscillator or the main oscillator and the synchronized oscillators are of a type in which the basic frequency setting is determined by an applied control magnitude, for example a control voltage, the whole instrument can be tuned in a very simple manner by adjusting the said voltage. This can also be utilized for making it possible to rapidly re-tune the instrument to another basic tone.

The invention is illustrated by means of example in the accompanying drawing, in which:

FIG. 1 shows a complete circuit diagram of an oscillator with automatic frequency control according to the invention.

FIG. 2 shows a number of time diagrams for illustrating the function of the oscillator circuit and,

FIG. 3 shows a schematic diagram for illustrating the interconnection of the different oscillator circuits.

An instrument comprises 12 oscillators which are all assumed to be of the embodiment shown in FIG. 1. One of the oscillators, in the given example that one which generates the lowest tone, however, differs from the remaining oscillators in that it has no automatic frequency control.

In the circuit shown in FIG. 1 the oscillator part is designated 05C The oscillator consists of a Schmitttrigger stage ST which is connected in feedback from the output to the input through a RC-circuit consisting of three resistances R R R and a capacitor C The input voltage to the Schmitt-trigger stage is formed by the voltage across the said capacitor C The stage ST can in a way known per se be switched between two positions in which it delivers two different voltages at its output. The output voltage is assumed to vary between -6 volt and 0 volt. The stage is triggered from one position to the opposite and vice versa at certain predetermined levels of the input voltage, i.e. the voltage across the capacitor C The oscillator functions as follows.

During an interval when ST delivers 6 volt at its output the capacitor C will be charged by a negative voltage from the output of the stage, i.e. the voltage across C decreases. At a certain value of the voltage of the capacitor the triggering level is reached and the stage switched to the opposite position, in which it delivers 0 volt at its output. The voltage across the capacitor C will now increase until the input voltage reaches the triggering level for switching the stage in opposite direction; then the capacitor voltage will again decrease and the procedure is repeated.

The signal from the stage ST will be a rectangular wave,

the period time of which, i.e. frequency of the said rectangular Wave, will in first hand be dependent upon the time constant of the feed back circuit. Of the resistances included in the feed back circuit the resistance R is manually adjustable. The value of capacitor C is different in different oscillators in order to allow adjustment of the oscillators to different tones in the octave.

The charging and discharging of the capacitor C and consequently the period time or frequency of the generated rectangular wave is also influenced by a positive voltage V fed to the capacitor through a resistance R; and a negative voltage which is applied to the capacitor through a manually adjustable potentiometer P. The voltage V is accurately stabilized and common for all oscillators. By varying the voltage V it is possible to displace the frequency of all oscillators, i.e. the tone scale of the whole instrument. The potentiometer P has for its purpose to compensate for the somewhat different influence upon the basic frequency setting, which a change of the voltage V otherwise should give rise to in different oscillators. The potentiometer P is consequently set individually in each oscillator such that the said voltage change will produce substantially the same relative frequency change in all oscillators. Instead of using the shown potentiometer arrangement a corresponding adjustment possibility can be obtained for example by making the resistance R; variable.

The basic frequency setting of the oscillator is in the given example determined by the setting of resistance R the value of voltage V and the setting of potentiometer P. Besides this there is in all oscillators except that one for generating the lowest tone f an automatic frequency control, which is produced by applying a control voltage to a terminal a',,. The control voltage is derived from another oscillator as described more in detail in the following.

As the two half periods of the output voltage from the stage ST are not equal in length and furthermore will vary mutually at variation of the total period time, the oscillator part also includes a bistable flip-flop FF which is triggered by one flank of the voltage from ST. The circuit FF produces frequency division by two of the voltage from the stage ST and delivers a rectangular voltage, in which the two half-periods are exactly equal in length. The voltage from PE is assumed to vary, likewise as the voltage from similar circuits, between 0 volt and 6 volt.

The rectangular voltage from FF forms the output voltage of the oscillator and is fed to an output terminal A A tone having a required acoustic colour is then generated from the voltage upon terminal A According to the foregoing a substantially chromatic scale will be obtained if the different oscillator circuits are regulated such that the mutual frequency relationship given by Equation 1 is fulfilled. The Equation 1 is rewritten as follows:

The control voltage is in accordance herewith generated by comparing 1/20 of the frequency of one oscillator with the difference between the frequencies of two other oscillators, more particularly those two oscillators which are a whole tone step (two half tone steps), and three half tone steps respectively below the oscillator frequency in question.

Generation of the frequency f /ZO is in the shown example effected in three steps, first division of the oscillator frequency f,, by two in a first bistable flip-flop FF then further frequency division by two in a similar flipflop FF and finally division of the obtained frequency by five in a dividing circuit Sc which said last circuit can be composed in a way known per se by three bistable flip-flops. The voltage from the first flip-flop FF is led to an output terminal B and is used for generating the corresponding tone in adjacent lower octave. The voltage from the second flip-flop FF is led to a terminal C 5 and is used for generating the corresponding tone in the next following lower octave.

The frequency f /ZO obtained by the frequency division is compared with the frequencies f,, f in a coincidence unit or and-gate consisting of diodes D D D D The inputs to the coincidence unit b,,, a are in accordance herewith connected to the output of those osci lators which are three half and one whole tone step below the frequency of the oscillator in question. The coincidence current produced by the coincidence unit is integrated in a capacitor C which delivers the control voltage for the automatic frequency control. The control voltage is derived from an output terminal D,,. The control voltage appearing at the output D is used for regulating one of the input frequencies to the coincidence unit, in the given example the frequency f and the said control output is therefore connected to the control input d of that oscillator which is three half tone steps below the frequency of the shown oscillator. The control input d of the shown circuit is correspondingly connected to the output D of that oscillator which is three half tone steps above the frequency of the shown oscillator.

The coincidence unit operates as follows.

The voltage at all inputs of the unit varies according to the foregoing between and -6 volt in a rhythm determined by the actual frequencies. As long as any of the inputs has the voltage 0 the common point P for the diodes D -D will be maintained at the potential 0 and the diode D is cut off, as the capacitor C has a negative potential. Only during those intervals when all inputs to the coincidence unit has 6 volt can the potential of point P fall below the value 0, whereby the diode D is opened and the capacitor C charged negatively from a voltage source 6 volt. During intermediate intervals when coincidence does not occur the capacitor C will be discharged.

The voltage pulses from $0 representing the frequency i divided by 20 can be regarded as opening pulses or gate pulses to the coincidence unit. Decisive for how much the capacitor C will be charged and consequently for the mean voltage value across the capacitor, i.e. the control voltage, therefore will be the total time during which negative voltage pulses appear simultaneously at inputs a and b during the said gate pulses.

If no special measures were taken an appreciable discharging of the capacitor C should take place in the intervals between two gate pulses resulting in a pronounced saw toot-h shape of the voltage across the capacitor C In order to take care of this an inverted gate pulse is furthermore derived from Sc i.e. a voltage which is 0 during the gate pulse and negative during intermediate intervals, which voltage is applied to the capacitor C through a resistance R By suitable choice of the circuit constants it is hereby possible to eliminate the said tendency to discharging of the capacitor C during the intervals between the gate pulses, so that a practically even voltage across the capacitor C will be obtained apart from the small variations caused by the charging pulses from the coincidence unit.

The function is illustrated in FIG. 2 where the upper diagram shows the voltage at input b and the next diagram shows the voltage at input a The two voltages are according to the foregoing derived from two adjacent oscillators and will consequently differ in frequency a half tone step from each other. The following diagram shows coincidence between the pulses at input a and the pulses at input b Coincidence maximum occurs at t and t and coincidence minimum occurs at t The period time between two coincidence maxima (or minima) represents the beat frequency f -f which according to the foregoing is to be regulated to equality with the frequency of the actual oscillator divided by 20. This frequency is represented by the said gate pulses, which are shown in the following diagram. The last diagram in FIG. 2 shows by negative voltage pulses those intervals when negative voltage appears simultaneously at all inputs to the coincidence unit, so called tripple coincidence. -It is during these intervals charging of the capacitor C takes place and the total area of the shadowed surfaces in FIG. 2 will determine the charging condition of the capacitor and consequently the control voltage.

It is evident from the said figures that maximum coincidence ourrent and consequently maximum negative charging of the capacitor is obtained if the gate pulse occurs at moment t and the frequencies f and f are in the same phase. Should the gate pulse occur at moment t when the phase positions are opposite the coincidence current and hence the charging of the capacitor will be minimum. Assume that the gate is opened in the interval between t and t as shown in the drawing. In respect of the shown regulation circuit the frequencies f and f and consequently also f can be regarded as fixed frequencies. The frequency f is adjusted man'ually so that the relation f =f f is approximately fulfilled. If the relation were exactly fulfilled the gate pulse would remain in constant phase relative to the beat frequency represented by the said coincidence maxima and minima. If, however, the frequency f,, is for example a little too high the beat period is longer than the gate period which means that the gate pulse is gradually displaced towards t As a consequence the coincidence current increases and the capacitor C is charged to a higher negative voltage. Through the resistance R regulation current is delivered to the terminal d of the f oscillator thereby decreasing its frequency. By this negative feed-back the frequency f is automatically regulated to correct value and the relative phase is locked in the position where the feed-back current has the appropriate value. Stable synchronization is obtained as long as the negative gate signal is locked somewhere between the positions t and t By choosing a strong feed-back, for example by decreasing the resistance R the synchronization interval can be made very wide. A strong feed-back, however, may result in frequency modulation due to the fact that the voltage across capacitor C is not a perfect DC. voltage but is subject to small variations.

Instead of regulating the frequency f as in the given example it is of course also possible to choose the frequencies f and f fixed and used the feed-back signal for automatic control of the frequency f The interconnection between the different oscillators is illustrated in FIG. 3, where the references 1, 3 12 designate the number of the oscillator or tone in the octave and A, B, C, a, b, D and d designate the terminals in accordance with FIG. 1. The input signal fed in at b and a shall according to the given relation be a frequency which is three half and two half tone steps respectively below the frequency of the actual oscillator. For oscillator 1 this means that to its terminals b and a are to be applied frequencies corresponding to the tones 10 and 11 in the adjacent lower octave. But according to the foregoing this octave -is obtained from the output terminals B. The said terminals 12 and a are consequently connected to terminals B and B The next oscillator 2 is likewise to be fed with frequencies corresponding to tones in the adjacent lower octave and has its terminals b and a connected to the terminals B B To the oscillator 3 is to be applied a frequency corresponding to the highest tone in the adjacent lower octave and the basic tone in the actual octave. The said oscillator has consequently its terminals b a connected to B and A respectively. The remaining oscillators are to be fed at terminals a and b with frequencies corresponding to tones in the own octave and are connected to the outputs A: thus b a are connected to A and A respectively, b a are connected to A and A respectively etc.

As regards the control inputs d it is shown in the draw ing that the first oscillator has its terminal d unconnected. The oscillator 1 consequently operates with a set fixed frequency, which will determine the whole tone scale.

The remaining oscillators receive control voltage at input d from the output D of that oscillator which is three half tone steps higher in frequency, thus (1 is connected to D d is connected to D etc. The three last oscillators 10, 11 and 12 which should properly receive control voltage from the three first oscillators in the next higher octave, are however instead connected to the first oscillators in the actual octave as the oscillations from these also can be regarded as representative for the tones in the higher octave.

The basic frequency setting of each oscillator is as mentioned eifectuated by means of the manually variable resistance R Due to the described synchronization be tween the oscillators it is, however, not necessary at the frequency setting, except for the first oscillator frequency h. to judge if the actual tone height is correct or not but only that the basic setting is such that the synchronization is effective, i.e. that the oscillator setting is within the synchronization interval. This is easily proved in that, \when the synchronization limit is exceeded, the tone height will vary with a certain periodicity. The setting is effected such that the resistance R is first varied in one direction and thereafter in the opposite direction until the synchronization limit is in both cases exceeded and the tone is subject to fluctuations. The resistance is thereafter set halfway between the two obtained limit values. It is then ensured that the oscillator is adjusted to the centre of the synchronization interval.

The adjustment of the whole instrument should be carried through in a predetermined order determined by how the interconnection of the oscillators is made. In the given example the adjustment can be carried through as follows.

The frequency 1, of the first oscillator is assumed to be fixed and set to correct value. The frequencies f and i of the oscillators 2 and 3 are adjusted approximately as near as possible to their respective correct values and are regarded for the moment as constant. As is evident from FIG. 3 the oscillators 1, 3 and 12 are included in a closed regulation circuit in that they are interconnected such that the outputs A B are connected to the inputs a and b while the control output D is connected to the control input d The two given frequencies 12, and f thus determine the frequency ;f in the described regulation circuit. With the given values of f and f;, the oscillator 12 is adjusted to the centre of its synchronization interval. The oscillators for frequencies f f and i are included in a similar regulation circuit, in which the frequency f is determined by the given frequencies f and i The oscillator for frequency i is adjusted to the centre of its synchronization interval. Corresponding adjustments are then repeated in successive order for remaining oscillators. The adjustment procedure is illustrated in the following table.

Table 3 shows that the two last adjustments will produce a correction of the initially assumed values for f and f If the initial approximation for f and f was not sufficiently accurate it may be necessary to carry through the adjustment cycle once more with the new values on f and f The procedure converges so rapidly, however, that one tuning cycle is usually sufficient.

As shown in FIG. 2 only three or four coincidence pulses are obtained during a gate pulse. It is then evident that the integrated coincidence current in some degree will be dependent upon the position of these few coincidence pulses during the gate pulse interval. But the coincidence pulses can occur in any position during the interval and may change position from one charging interval to the next one. Hereby small variations in the voltage across the capacitor can arise, so called tripple coincidence fluctuations, which, if the feed-back is strong, can cause frequency modulation of the oscillators. In order to decrease this effect the gate pulse is to be made longer than shown in FIG. 2. A suitable value of the gate pulse interval is about 40% of the total period time.

In order to achieve a more smooth regulation of the oscillator, it is also possible to replace the shown comparison device operating according to the pulse coincidence principle with a phase sensitive detector, to which is applied on the one hand a voltage representing the oscillator frequency divided by 20 and on the other hand a voltage representing the said beat frequency, which said last voltage has been generated by super-imposing the actual frequencies and separation of the beat frequency by means of suitable filter circuits. The phase sensitive detector delivers as known an output signal which indicates as to size and sign the deviation in phase between the two compared voltages, which output signal can be utilized for regulation of the oscillator frequency in the same way as described.

It will be clear that it is equally possible to use the equation fix-12 10 and to suitably modify the interconnections between the oscillators.

A number of other modifications are also possible within the scope of the invention. Thus, in principle, frequency multiplication can be used instead of frequency division. In order to avoid frequency drift the independent oscillator or main oscillator may be crystal-controlled. Alternatively it is possible to use two main oscillators, one crystal-controlled and the other one adjustable, which oscillators are switched in as desired. The described synchronization method can in an analogous way also be applied to a mechanical system, for example rotating tone wheels. The difference between adjacent rotation frequencies can then be obtained by means of differential gears and multiplication or division is produced by means of common gear devices.

What is claimed is:

1. A device for generating an equal tempered twelve tone scale in electronic musical instruments, comprising twelve adjustable oscillators each having a basic frequen cy setting corresponding approximately to a different tone of an equal tempered scale, means for subtracting the frequencies of predetermined pairs of the oscillators, means for dividing the frequency of each of the oscillators by an integer m, means for comparing the result of subtracting each pair of frequencies with a predetermined one of the results of frequency division, means for generating a correction signal proportional to each comparison, and means for applying each correction signal to at least one of the oscillators from which the correction signal is derived.

2. A device as claimed in claim 1, wherein the frequency dividing means is a device for division of each oscillator frequency by 20, wherein the frequency thus obtained is compared with the beat frequency between the 11 minus second and n minus third oscillator frequencies and wherein a control signal produced by the said comparison is applied to one of the oscillators for maintaining the said relationship between the frequencies by negative feed-back in the correction circuit thus formed.

3. A device as claimed in claim 2, wherein the frequen cy division by 20 comprises two divisions by two in separate circuits, and wherein the frequencies obtained by the said divisions by two are utilized for generating the corresponding tones in the adjacent lower octaves.

4. A device as claimed in claim 1, the oscillators are of a type in which the basic frequency setting is determined by an applied control magnitude, for example a control voltage, further comprising means for applying a control magnitude to all oscillators in order to make possi ble a change of the basic frequency setting in all oscillators by a variation of the said control magnitude.

5. A method for generating an equal-tempered tone scale in electronic musical instruments, comprising the steps of generating an approximation of each tone in an equal-tempered scale, dividing the frequency of each generated tone by an integer m, subtracting the frequencies of predetermined pairs of the generated tones, comparing the result of each frequency division with the result of the subtraction of the predetermined tone pairs, generating a control signal proportional to each comparison, and adjusting the frequency of at least one of each of the tones corresponding to each comparison by the control signal generated by that comparison, wherein at least one of the tone generators is not adjustable.

6. A method as claimed in claim 5, wherein the scale has n tones, wherein the integer m is equal to 20, wherein the predetermined tone pairs are the n minus second and n minus third tones of the scale, and wherein the result of each subtraction is compared with /30 of the n tone.

7. A method as claimed in claim 5, wherein the scale has n tones, wherein the integer m is equal to 30, wherein the predetermined tone pairs .are the n minus second and 11 minus third tones of the scale, and wherein the result of each subtraction is compared with A of the 11+ seventh tone of the scale.

8. A method for generating an equal tempered tone scale in electronic musical instruments, comprising the steps of generating an approximation of each tone in an equal tempered scale, subtracting predetermined pairs of the generated tones multiplying the result of each subtraction by an integer m, comparing the results of each multiplication with a predetermined tone different from the corresponding subtracted tones, generating a control signal proportional to the result of each comparison, and adjusting the frequency of at least one of the tones involved in a comparison with a corresponding control signal, wherein at least one of the tone generators is not adjustable.

9. A method as claimed in claim 8, wherein the scale has n tones, wherein the integer in is equal to 20, wherein the predetermined tone pairs are the 11 minus second and n minus third tones of the scale, and wherein the result of each subtraction is compared with twenty times the nth tone of the scale.

10. A method as claimed in claim 8, wherein the scale has n tones, wherein the integer m is equal to 30, wherein the predetermined tone pairs are the n minus second and 11 minus third tones of the scale, and wherein the result of each subtraction is compared with thirty times the it plus seventh tone of the scale.

References Cited UNITED STATES PATENTS 2,293,499 8/ 1942 Fisher 84-1.01 2,864,956 12/1958 Makow 331-2X 2,987,680 6/1961 Israel 33 l-2 3,202,930 8/1965 Muraszko 331-2 OTHER REFERENCES Culver: Musical Acoustics, McGraW-Hill, 1956, ML

Handbook of Chemistry and Physics, mathematical tables from, 8th edition, 1947, p. 245.

HERMAN K. SAALBACH, Primary Examiner W. N. PUNTER, Assistant Examiner US. 01. 3, 331 2, 37, st

UNITED STATES PATENT OFFICE PO-lOSU (5/ 9) 6 CERTIFICATE OF CORRECTION Patent No. 3 520,982 Dated Julv 21,. 1970 lnventm-(s) KARL.G. MALMFORS It is certified that error appears in the aboveidentified patent and that said Letters Patent are hereby corrected as shown below:

Col 8, line 17, after "the" (first occurence) insert --two-,-

Cole 8, line 30,, in the equation ":En-l" should be --fn-2-; Signed and sealed this 11th day of April 1972 (SEAL) Attest:

EDWARD M FLETCHER,JR Attesting Officer ROBERT GOTTSCHALK Commissioner of Patents 

